Ideas for Capstone or Honors Projects in Mathematics
The ideas listed below for honors projects may spark an idea for a project. They will also give you some ideas about what certain faculty members are interested in.
Students are also invited to offer their own ideas for projects based on their own reading, coursework, or perhaps based on earlier work (for example, in a summer REU). In that case, you should feel free approach a faculty member who might be willing to work with you. (Or contact Ron Freiwald for suggestions about whom to talk with.)
Some of the ideas listed below are harder and some easier. Some involve working on actual problems while others involve learning about a problem and why it's important or interesting. In some cases, there may be an easier version (special case) of a problem that is more accessible. If one of the areas sounds interesting to you, contact the faculty member to discuss the topic and your background in more depth.Math majors with a special interest in sciences should also explore the ideas found on the webpages for Biology, Chemistry, Earth and Planetary Sciences, and Physics (click on the marker to "Research for Undergrads" in the left frame). If there are ideas that involve also work with mathematics there, we're certainly will to explore some sort of cooperative project with you.
Professor Al Baernstein (Analysis)I can direct a project involving random walks or related stochastic processes.
Professor Renato Feres (Geometry)Some of following problems are meant to introduce you to advanced but well-established topics in graduate level math, others are much more open ended and may lead to original results. Some are more "theoretical" while others invite you to do some computer exploration. A few are probably pie-in-the-sky problems that, to me at least, are amusing to contemplate. Whatever the case, I'd be happy to discuss any of them with you and suggest reading material for anything in this list that strikes your fancy.1) The kinematics of rolling. (Riemannian geometry/Non-holonomic mechanical systems)Professor Steve Krantz (Analysis, Complex Variables, Geometry)
On a smooth stone, draw a curve beginning at a point p, and hold the stone over a flat table with p as the point of contact. Now roll the stone over the plane of the table so that at all times the point of contact lies on the curve, being careful not to allow the stone to slip or twist. We may equally well think that we are rolling the plane of the table over the surface of the stone along the given curve. Mechanical systems with this type of motion are said to have "non-holonomic" constraints, and are common fare in mechanics textbooks.
Now imagine a tangent vector to the plane at p. This rolling of the plane over the surface provides a way to transport v along the curve, keeping it tangent at all times. The resulting vector field over the curve is said to be a "parallel" vector field. Show that there is a unique way to carry out this parallel translation. (Find a differential equation that describes the parallel vector field and use some appropriate existence and uniqueness theorem.) Let c be a short path joining p and q, whose velocity vector field is parallel. Show that c is the shortest path contained in the surface that joins p and q.
Whether or not you fully succeed, this mechanical idea will give you a concrete way of thinking about ideas in differential geometry that might seem a bit abstract at first, such as Levi-Civita connection, parallel translation, geodesics, etc. Also look for an engineering text on Robotic manipulators and explain why such non-holonomic mechanical systems are important in that area of engineering.
I don't know of many places where these things are explained in a simple way. Perhaps Geometric Control Theory by Velimir Jurdjevic is a place to start. In the engineering literature, "A mathematical introduction to Robotic Manipulation" is a particularly good reference.
2) Geometry in very high dimensions. (Convex geometry)
Geometry in very high dimensions is full of surprises. Consider the following easy exercise as a warm-up. Let B(n,r ) represent the ball of radius r, centered at the origin, in Euclidian n-space. Show that for arbitrarily small positive numbers a and b, there is a big enough N such that (100 - a)% of the volume of B(n,r ) is contained in the shell B(n,r ) - B(n,r - b ) for all n > N.
Here is a much more surprising fact that you might like to think about. Let S(n-1) denote the sphere of radius 1 in dimension n. (It is the boundary of B(n,1 ).) Let f be a continuous function from S(n-1) into the real line that does not increase distances, that is, | f(p) - f(q) | is not bigger than | p - q | for any two points p and q on the sphere. ( f is said to be a "1-Lipschitz" function.) Then there exists a number M such that, for all positive a, no matter how small, the set of points p in S(n-1) such that | f(p) - M |>a has volume smaller than exp(-na^2 / 2 ). In words, this means that, taking away a set with very small volume (if the dimension is very large), f is very nearly a constant function, equal to M.
This is much more than a geometric curiosity. In fact, such concentration of volume phenomenon is at the heart of statistics, for example. To make the point, consider the following. Let S(n-1, n^0.5) be the sphere in n-space whose radius is the square root of n. Let f denote the orthogonal projection from the sphere to one of the n coordinate directions, which we agree to call the x-direction. Show that the part of the sphere that projects to an interval a < x < b has volume very nearly (when n is big) equal to the integral from a to b of the standard normal distribution. (This is easy to show if you use the central limit theorem).
For a nice introduction to this whole subject, see the article by Keith M. Ball in the volume Flavors of Geometry, Cambridge University Press, Ed.: S. Levy, 1997.
3) Hodge theory and Electromagnetism. (Algebraic topology/Physics)
Electromagnetic theory since the time of Maxwell has been an important source of new mathematics. This is particularly true for topology, specially for what is called "algebraic topology". One fundamental topic in algebraic topology with strong ties to electromagnetism is the so called "Hodge-de Rham theory". Although in its general form this is a difficult and technical topic, it is possible to go a long way into the subject with only Math 233. The article "Vector Calculus and the Topology of Domains in 3-Space", by Cantarella, DeTurck and Gluck (The American Mathematical Monthly, V. 109, N. 5, 409-442) is the ideal reference for a project in this area. (It has as well some inspiring pictures.)
Another direction to explore is the theory of direct current electric circuits (remember Kirkhoff's laws?). In fact, an electric circuit may be regarded as electric and magnetic field over a region in 3-space that is very nearly one dimensional, typically with very complicated topology (a graph). Solving circuit problems implicitly involve the kind of algebraic topology related to Hodge theory. (Hermann Weyl may have been the first to look into electric circuits from this point of view.) The simplification here is that the mathematics involved reduces to finite dimensional linear algebra. A nice reference for this is appendix B of The Geometry of Physics (T. Frankel), as well as "A Course in Mathematics for Students
of Physics" vol. 2, by Bamberg and Sternberg.
4) Symmetries of differential equations. (Lie groups, Lie algebras/Differential equations)
Most of the time spent in courses on ODEs, like Math 217, is devoted to linear differential equations, although a few examples of non-linear equations are also mentioned, only to be quickly dismissed as odd cases that cannot be approached by any general method for finding solutions. (One good and important example is the Riccati equation.) It turns out that there is a powerful general method to analyze nonlinear equations that sometimes allows you to obtain explicit solutions. The method is based on looking first for all the (infinitesimal) symmetries of the differential equation. (A symmetry of a differential equation is a transformation that sends solutions to solutions. An infinitesimal symmetry is a vector field that generates a flow of symmetries.) The key point is that finding infinitesimal symmetries amounts to solving linear differential equations and may be a much easier problem than to solve the equation we started with.
Use this idea to solve the Riccati equation. Choose your favorite non-linear differential equation and study its algebra of infinitesimal symmetries (a Lie algebra). What kind of information do they provide about the solutions of the equation? Since my description here is hopelessly vague, you might like to browse Symmetry Methods for Differential Equations - A Beginner's Guide by Peter Hydon, Cambridge University Press. It will give you a good idea of what this is all about.
5) Riemann surfaces and optical metric. (Riemannian geometry/Optics)
Light propagates in a transparent medium with velocity c/n, where c is a constant and n is the so called "refractive index" -- a quantity that can vary from point to point depending on the electric and magnetic properties of the medium. For a given curve in space, the time an imaginary particle would take to traverse its length, having at each point the same speed light would have there, is called the "optical length" of the curve. Therefore, the optical length is the line integral of n/c along the curve with respect to the arc-length parameter. According to Fermat's principle, the actual path taken by a light ray in space locally minimizes the "optical length". It is possible to use the optical length (for some given function n) to defined a new geometry whose geodesic curves are the paths taken by light rays. This is a particular type of Riemannian geometry, called "conformally" Euclidian. All this also makes sense in dimension 2.
One of the most famous paintings of Escher show a disc filled with little angels and demons crowding towards the boundary circle. What refractive index would produce the metric distortions shown in that picture?
A fundamental result about the geometry of surfaces states that, no matter what shape they have, you can always find a coordinate system in a neighborhood of any point that makes the surface conformally Euclidian. Why is this so? (This will require that you learn something about so called "isothermal coordinates".)
6) Random walks and diffusion limits (I). (Probability theory/ElementaryGeometry)
Imagine a long and narrow cylinder of radius r and a point particle that moves in the region bounded by the cylinder. The motion is specified as follows: starting at a point on the inner wall of the cylinder, choose at random a direction and let the particle move with constant speeduntil it hits another point of the cylinder. Once there, choose a new direction at random and repeat the process. A natural scheme (for reasons I won't describe here) is to pick the random direction with probability proportional to the cosine of the angle it makes with the (inward pointing) normal vector. The problem is to determine the probability that the particle will be given distance away from the initial point at a given time in the future. It is actually hard to find such a probability explicitly, but if the cylinder is very narrow and the particle moves very fast (with speed proportional to the reciprocal of the radius) you can use the central limit theorem to obtain an explicit (Gaussian) approximation. What is the variance of the resulting normal law? How does the variance change if the cross section of the tube is, say a square, instead of a circle?
7) Random walks and diffusion limits (II). (Probability theory)
We can, of course, consider a two dimensional variant of the previous problem, in which the cylinder consists of two infinite parallel lines and the particle velocity after collision is chosen according to the same cosine law. However, after some thought you will realize that the hypothesis of the central limit theorem fail (barely!) to hold. Nevertheless, we can still ask what kind of limit process this random walk leads to. (Some key words: stable distributions, Levy processes.)
8) Random Billiards. (Billiard systems/Probability theory)
You may have heard a lot about the mathematical theory of chaos. It is part of the general subject of Dynamical Systems. In the tool box of the practitioners of this subject is a kind of toy system that is used to explore and illustrate almost any conceivable dynamical behavior (including chaos), called "billiard systems". It is just what you might expect: a billiard table and a point mass that moves about and bounces off the sides according to the law of mirror reflection. But the table is allowed any shape you want.
The problem I would like to propose is actually related to 6) and 7). Take the setting of RW2, except that the two parallel lines, when examined with strong lenses, reveal a periodic structure. More precisely, replace those lines with the graphs of, say, C sin(x / C) and 1+ C sin(x / C), where C is very small. Intuitively, as C approaches 0, the (deterministic) billiard system should behave more and more like the probabilistic system of 7). How can this intuition be made precise? What kind of scattering probability results after passing to the limit? What does the cosine law of 6) and 7), in particular, have to do with all this?
9) Existence of surfaces. (Computer Science/Differential Geometry)
You've probably heard of cellular =automata. The most celebrated example among them is John Conway's "game of life". They are, in general, a sort of beads game played over an infinite lattice (grid), which in our case will have dimension 3. At each moment, a lattice point may be empty or occupied by a bead of one among a number of colors. At the next moment, the state of that lattice point is renewed according to some function of the state of the nearest neighbor points. This
function specifies the rules of the game.
Our problem is to find rules that will cause the beads to organize themselves into "surfaces". (Suggestion: try to find rules that imitate the behavior of amphiphilic molecules, like
the lipid bilayers that make up biological membranes. These molecules have one end that "likes" water and another that "hates" it.) If such surfaces can be obtained, is it possible to control how "crumpled" or "smooth" they are? or to control their curvature? Is it possible to make sense of notions such as differentiability and curvature in this discrete setting? (This would require the passage to some appropriate scaling limit.)
10) Chemical varieties. (Algebraic geometry/Chemical kinetics)
Algebraic geometry studies the geometry of sets of solutions of systems of polynomial equations (typically over the field of complex numbers) and how that geometry relates to the algebra of all polynomials that vanish on the set.
It is not difficult to show that to every system of chemical reactions with specified reaction speeds is associated a system of nonlinear first order differential equations describing how reactant concentrations change in time. These differential equations are of a very special kind: on the left-hand side is the first derivative of each reactant concentration (in moles) and on the right a polynomial function of the concentrations, whose coefficients are the stoechiometric constants. (Incidentally, the whole business of stoechiometry and its linear algebra underpinnings is in itself a great subject for a project.)
The set of zeros of the polynomial equation are equilibrium concentrations for the chemical reactions. Call the set of complex solutions of the polynomial equations the "Chemical Variety" of the system of reactions. These should be very special algebraic varieties. (They are typically of degree 2, for example, for any reasonable reaction mechanism.) Choose your favorite reaction mechanism and describe, in as much detail as you can, the geometric properties of the associated chemical variety. Are there interesting special properties shared by all chemical varieties?1) It is known that, given any closed planar curve, there are four points on that curve which are the corners of a rectangle. It is an open problem whether there will be four points that are the corners of a square.
2) Let U be a planar region, and let G be the group of rigid motions of the plane that map U to itself. We call G the "automorphism group" of U , and we denote it by Aut( U ). Now suppose that U' is a small perturbation of U . How is Aut( U' ), as a group, related to Aut( U ) ? How does the answer change as U' deviates farther and farther from U ?
3) (Refer to (2) for terminology.) Let G be any finite group. Is there a planar domain U such that Aut( U ) = G ? Can we relate the topology of U to the structure of the group? What if we allow U to live in a higher dimensional space? Does that allow more groups G to give an affirmative answer? Given a group G, can we estimate the dimension of the space in which a domain U will live that has the desired property?
4) (Refer to (2) for terminology.) It is an intuitively obvious assertion that, of all planar domains, the disc has the "largest" automorphism group. Formulate a precise version of this statement and prove it. Given any group G that is the automorphism group of some planar domain, can we find a particular planar domain U that is as close to the disc as we please and so that Aut( U ) = G ?
5) Consider the space C^\infty of infinitely differentiable functions and the space C^\omega of real analytic functions (i.e., functions with convergent power series expansions). Of course C^\omega is a subset of C^\infty. Is there a range of function spaces, perhaps a range that is parametrized, that spans the gamut between C^\omega to C^\infty ? (This problem is important for the theory of partial differential equations.)6) Let U be a convex planar domain. Call a point p in U an equichordal point if all chords of U that pass through p have the same length. It is known that a convex planar U can have at most one equichordal point. But the proof is very abstract and extremely difficult. Problem 1: find an elementary proof. Problem 2: What is true in dimension three? Problem 3: What is true for non-convex domains?
Professor Mohan Kumar (Algebra)1) If a1,a2,...an are integers with gcd = 1, then the Eulidean algorithm implies that there exists a (n x n)-matrix A with integer entries, with first row = (a1,a2,...,an), and such that det(A) = 1. A similar question was raised by J.P. Serre for polynomial rings over a field, with the a's being polynomials in several variables. This fundamental question generated an enormous amount of mathematics (giving birth to some new fields) and was finally settled almost simultaneously by D. Quillen and A. A. Suslin, independently. Now, there are fairly elementary proofs of this which require only some knowledge of polynomials and a good background in linear algebra. This could be an excellent project for someone who wants to learn some important and interesting mathematics. (These results seem to be of great interest to people working in control theory. Though I am not an expert, I'm willing to learn with a motivated student.Professor John McCarthy (Analysis)
2) A basic question in number theory and theoretical computer science is to find a `nice' algorithm to decide whether a given number is prime or not. This has important applications in secure transmissions over the internet and techniques like RSA cryptosystems. Of course, the ancient method of Eratosthenes (sieve method) is one such algorithm, albeit a very inefficient one. All the methods availabe so far has been known to take exponential time. There are probabilistic methods to determine whether a number is prime, which take only polynomial time. The drawback is that there is a small chance of error in these methods. So, computer scientists have been trying for the last decade to find a deterministic algorithm which works in polynomial time. Recently, this has been achieved by three scientists from IIT, Kanpur, India. A copy of their article can be downloaded from www.cse.iitk.ac.in
A nice project would be to understand their arguments (which is very elementary and uses only a little bit of algebra and number theory) and may be to do a project on the history of the problem and its ramifications.1) (Fluid Dynamics) Consider a cylindrical tube, open at one end. At the closed end, a small quantity of gas is injected. It diffuses out the other end at a predictable rate. Now, suppose the quantity of gas injected is increased. The flow will not scale linearly, as the effect of the pressure of the introduced gas must be considered. I have a project with Professor Gregory Yablonksky in the Chemical Engineering department to model this flow.Professor Stanley Sawyer (Probability, Bology, Scientific Computing)
2) (Linear Matrix Inequalities) A computer vision problem posed by Professor Robert Pless in the Computer Science Department. Imagine a large number of cameras arranged around a central object. One wants to match up the pictures, but there is some error in the measurement. Mathematically, the problem becomes approximating a large symmetric matrix by a rank 3 matrix that has 1's on the diagonal. It ties in to an active research area in systems theory: solving a linear matrix inequality with a rank constraint. Nobody knows how to do this well.
3) (Applied Statistics/Public Health) The "French paradox" is the claim that, despite having a high fat diet, French people have a low rate of heart disease. I believe this is a statistical artifact, due principally to cultural differences in filling out death certificates. I would be willing to supervise an undergraduate who wished to hunt down the data and analyze it.Mathematical population genetics: What determines the fate of a gene in a population? How can good genes go extinct? How fast are genes (good or bad) lost? How important are random forces? How good are the approximations?Professor John Shareshian (Algebra, topological combinatorics)
The methods that will be used are Markov processes in probability theory, diffusion processes in probability theory, scientific computing to analyze problems that are difficult to analyze theoretically, or a combination of these.1) Computation in topological combinatorics - Topological combinatorics includes the study of simplicial complexes (that is, geometric objects built from possibly higher dimensional analogues of the unit interval, the equilateral triangle and the equilateral tetrahedron) whose faces are indexed by combinatorial objects such as graphs. The Homology program of J.-G. Dumas, F. Heckenbach, D. Saunders and V. Welker has been used to investigate the structure of such complexes. There are many adjustments and additions which could be made to improve the program, the most ambitious of which is to make it amenable to parallel processing.Professor Victor Wickerhauser (Applied and Computational Mathematics, Wavelets)
2) Order complexes of subgroup lattices - The set of subgroups of a group G is partially ordered by inclusion. There are interesting open questions and proven theorems about relating the algebraic structure of G to the combinatorial structure of this partially ordered set. For any partially ordered set P, the set of all totally ordered subsets of P determines a simplicial complex. The topological structure of this complex is related to the combinatorial structure of P. One can hope to use this relationship productively when P is the set of subgroups of G. This area is appropriate for both reseach and expository projects.
3) Symmetric functions - A symmetric function is a power series of bounded degree in infinitely many variables which is not changed by any permutation of the variables. Symmetric functions appear in many areas of mathematics, including combinatorics and representation theory (which involves studying a group G by understanding homomorphisms from G to various matrix groups). There are lots of interesting open combinatorial problems involving symmetric functions (many appear in the exercises after Chapter 7 of R. P. Stanley's book, Enumerative Combinatorics, Volume 2). This area is also appropriate for expository projects.1) Read Daubechies and Sweldens "Factoring Wavelet Transforms into Lifting Steps," (J. Fourier Anal. Appl. 4:3(1998),245-267). Then implement the Euclidean algorithm for Laurent polynomials described in the paper. (Thus, you will use ideas in abstract algebra and Fourier analysis to write an efficient computer program that is part of the JPEG-2000 image compression algorithm.)
2) Read chapter 3 (pp. 67-101) of my book "Adapted Wavelet Analysis," and also Strang, "The Discrete Cosine Transform" (SIAM Review 41:1(1999),135-147). Synthesize a proof that the discrete Hartley transform is orthogonal. (Thus, you will see how the Sturm-Liouville theorem from differential equations can save many tedious computations in the verification that a basis, such as one used in the JPEG (1990) image compression algorithm, is orthonormal.)
Professor Guido Weiss (Analysis)
A large number of undergraduate research projects can be obtained by studying various reproducing systems (of vectors or functions). Let us consider an example.
An orthonormal basis (in Euclidean space or, more generally, a Hilbert space) is such a "reproducing system" in the sense that an arbitrary vector, v, equals the linear combination obtained by multiplying the individual elements of the basis elements by their inner products with v and then summing the vectors obtained. Many such bases can be constructed by selecting an appropriate vector (or function) in the space being considered and applying certain basic operations on this function (translations, dilations, and modulations, for example). Wavelets are examples of such systems and their construction offers a wide variety of research projects.